Consider a smooth complex projective variety \(M\). To understand the group of birational transformations (resp. regular automorphisms) of \(M\), one can use tools from Hodge theory, dynamical systems, and geometric group theory. I shall try to describe several of these techniques by looking at one specific question: if a finite index subgroup of \(\mathrm{SL}(n,\mathbb Z)\) acts faithfully on \(M\) by birational transformations, is the dimension of \(M\) larger than or equal to \((n-1)\)?